In lecture we saw the Cournot competition model for two firms with the same cost function. Now, we are going to consider asymmetric cost functions. Assume that demand for a good is given by p=abQd (Qd is quantity demanded), and that there are 2 firms competing in quantities. Both have no fixed costs and a constant marginal cost. Firm 1 has a marginal cost c1, and firm 2 has a marginal cost c2. We have that a>c1>c2.

Find the reaction functions of firms 1 and 2 in this market: how the optimal quantity produced depends on the quantity produced by the other firm.

To verify that you have found the correct reaction functions, compute the optimal q1 if q2=100, a=4, b=0.01, c1=2, and c2=1. (Note that this is not necessarily an equilibrium.)

Q1= 50.

E1)Solve for the quantity produced by each firm and the equilibrium price.

To verify that you have found the correct equilibrium, compute q1, q2, and p if a=4, b=0.01, c1=2, and c2=1.

q1=?

q*2=?

p*=?

E2)Find the equilibrium price and the quantity produced by each firm if they compete in prices (Bertrand competition). (Assume the parameters given above.)

p is close to what value?

a)C1

b)None of the above

c)C2

d)C1/2+C2/2

e)0

E3)q1 is close to what value?

a)0

b)None of the above

c)a/b-C1/b-C2/b

d)a/b-C1/b

e)a/b-C1/b

E4)q2 is close to what value?

a)a/b-C1/b

b)a/b-C1/b-C2/b

c)a/b-C2/b

d)None of the above

e)0

E5)How does this equilibrium compare to the perfectly competitive case (if firms sold at their marginal cost as though they faced perfect competition)?

a)Bertrand competition results in an efficiency gain relative to perfect competition

b)Bertrand competition results in an efficiency loss relative to perfect competition